Valuation of Equities and Currencies Using the Black-Scholes Model
Some underlying instruments have carry benefits. These benefits include dividends for stock options, foreign interest rates for currency options, and coupon payments for bond options. The BSM model should be adjusted to incorporate carry benefits in the option value. Let…
Components of the BSM Model
The BSM model for pricing options on a non-dividend-paying stock is given by: European Call $$ c_0= S_0N(d_1) – e^{(-rT)}KN(d_2) $$ European Put $$ p_0=e^{-rT}KN\left({-d}_2\right)-S_0N\left({-d}_1\right) $$ Where: $$ \begin{align*} d_1 &=\frac{ln{\left(\frac{S}{K}\right)}+\left(r+\frac{1}{2}\sigma^2\right)T}{\sigma\sqrt T} \\ d_2 &=d_1-\sigma\sqrt T \end{align*} $$ \(N(x)\) =…
Assumptions of the Black-Scholes-Merton Option Valuation Model
The Black-Scholes-Merton (BSM) model is an optional pricing model. Under this model, the underlying share prices evolve in continuous time and are characterized at any point in time by a continuous distribution rather than a discrete distribution. The following key…
Expectations Valuation Approach
One-step Binomial Tree Since a hedged portfolio returns the risk-free rate, it can determine the initial value of a call or put. The expectations approach calculates the values of the option by taking the present value of the expected terminal…
Valuation of an Interest Rate Option
Interest rate options are options with an interest rate as the underlying. A call option on interest rates has a positive payoff when the current spot rate is greater than the exercise rate. $$ \begin{align*} \text{Call option payoff} & =\text{Notional…
Arbitrage Opportunities Involving Options
Call Option A hedging portfolio can be created by going long \(\phi\) units of the underlying asset and going short the call option such that the portfolio has an initial value of: $$ V_0=\phi S_0-c_0 $$ Where: \(S_0\) = The…
No-Arbitrage Values of Options
Valuing European Options A European option is an option that can only be exercised at expiry. Let’s consider a simple example to better illustrate the concept. Example: The Value of a European Option Consider a stock with an initial price…
Binomial Option Valuation Model
Contingent Claims A contingent claim is a derivative contract that gives the owner the right but not the obligation to receive a future payoff that depends on the value of the underlying asset. Call and put options are examples of…
Stock Value Based on FCF Valuation Model
If the per-share value of equity obtained from the model is lower than the share price, the stock is overvalued. If the per-share value of equity obtained from the model is higher than the share price, the stock is…
Approaches for Calculating the Terminal Value
The terminal value can be calculated using a: Single-stage (constant-growth) model. Valuation multiples approach. Under the valuation multiples approach, there are two ways this can be done. $$\begin{align*} & \text{Terminal value in year }n \\ &=\text{Justified trailing P⁄E}\times\text{Forecasted earnings…
